Did he kill.........

I don't think he did.

I"m guessing he didn't since she didn't wrong him in anyway.

I certainly hope he didn't, but unfortunately there is a possiblity he did, at that point he was already settling scores and my perspective on it is that he 'felt' he had a connection with her (because of the head blown thing), now at that point we also learn the previous scenes with her were a fantasy, and with becoming hostile with him she become on of the 'others'.

Nothing suggests it

The gun to head hand gesture followed by the daunting music does give that impression but it's open to interpretation I guess.

I actually watched a WatchMojo video (Sorry)...

It did suggest that he did kill her, and that was the first time I had thought of it.

Like the rest of the movie you don't know for sure what is real or not, but they do suggest that the scene after the 'reveal' we see Joker on his own and police sirens can be heard in the background

I’m frustrated by the lack of clarity on this topic as well.

Makes no sense for him to kill her, but let the midget go.

He also didn’t kill his former boss, who definitely wronged him.

Yeah he did